Recent advances in PET (positron emission tomography) instrumentation have greatly improved the utility of nuclear medicine for studying biochemical function and regional blood flow through high resolution imaging of in- vivo distributions of radionuclides. In most clinical settings, these systems use relatively simple image reconstruction schemes that are based on the well known filtered backprojection algorithm. Despite the computational simplicity and ease of analysis of these methods, they do not fully exploit the potential of PET as they fail to take account of the inherently photon limited nature of the data. Over the last decade, there has been considerable effort extended towards the development of PET image reconstruction methods that are based on more sophisticated statistical models and estimation schemes. Emphasis in this work has largely concentrated on qualitative assessment of the resulting images. It is the thesis of this project that the power of these methods lies in their ability to provide accurate quantitative information, particularly in dynamic studies or other situations where the number of detected photons is low. The purpose of this project is to develop and assess new statistical estimation schemes for PET image reconstruction and to map them onto a vector processor for fast computation. These methods will incorporate an accurate statistical model for the data that includes factors such as random and scatter events, deadtime, attenuation, detector response and crystal penetration and scatter. The formulation will follow a Bayesian paradigm, in which a statistical model, known as a Markov random field (MRF), will be used to represent the image. In addition to providing an attractive model for PET images, the MRF is a useful mechanism by which structural information, extracted from anatomical magnetic resonance (MR) scans, may optionally be incorporated into the reconstruction process. The methods will be tested using Monte Carlo simulation, phantom studies and human data. The Monte Carlo and phantom studies will concentrate on comparing quantitative accuracy (bias and variance) using filtered backprojection and Bayesian methods, with and without a priori anatomical information. It is anticipated that these new reconstruction methods will prove clinically useful on several levels. Improvements in quantitative accuracy should lead to enhanced contrast and hence improved detection of abnormalities in oncological studies. Quantitatively accurate PET will be better able to estimate absolute and relative uptake in anatomical regions of interest and hence provide improved monitoring of the progression of disease and the efficacy of therapeutic intervention. The statistical estimators will be particularly useful in situations where the number of detected events per frame is low, for example in whole body studies of metastatic disease, and pharmacokinetic applications as well as neuro-receptor and stimulated brain activation studies.